Paper 1, Section II, E

Analysis I | Part IA, 2014

(i) Prove Taylor's Theorem for a function f:RRf: \mathbb{R} \rightarrow \mathbb{R} differentiable nn times, in the following form: for every xRx \in \mathbb{R} there exists θ\theta with 0<θ<10<\theta<1 such that

f(x)=k=0n1f(k)(0)k!xk+f(n)(θx)n!xnf(x)=\sum_{k=0}^{n-1} \frac{f^{(k)}(0)}{k !} x^{k}+\frac{f^{(n)}(\theta x)}{n !} x^{n}

[You may assume Rolle's Theorem and the Mean Value Theorem; other results should be proved.]

(ii) The function f:RRf: \mathbb{R} \rightarrow \mathbb{R} is twice differentiable, and satisfies the differential equation ff=0f^{\prime \prime}-f=0 with f(0)=A,f(0)=Bf(0)=A, f^{\prime}(0)=B. Show that ff is infinitely differentiable. Write down its Taylor series at the origin, and prove that it converges to ff at every point. Hence or otherwise show that for any a,hRa, h \in \mathbb{R}, the series

k=0f(k)(a)k!hk\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k !} h^{k}

converges to f(a+h)f(a+h).

Typos? Please submit corrections to this page on GitHub.